Hilbert Functions of Filtered Modules

TL;DR

This paper reviews three decades of research on Hilbert functions of modules over local rings, emphasizing new developments, elementary proofs, and applications to graded algebras like fiber cones and Sally-modules.

Contribution

It unifies recent advances in Hilbert function theory for filtered modules with a focus on elementary techniques and applications to non-filtration graded algebras.

Findings

Extended classical Hilbert function results to filtered modules.

Presented elementary proofs using superficial elements.

Connected Hilbert functions of filtrations to graded algebras like fiber cones.

Abstract

In this presentation we shall deal with some aspects of the theory of Hilbert functions of modules over local rings, and we intend to guide the reader along one of the possible routes through the last three decades of progress in this area of dynamic mathematical activity. Motivated by the ever increasing interest in this field, our goal is to gather together many new developments of this theory into one place, and to present them using a unifying approach which gives self-contained and easier proofs. In this text we shall discuss many results by different authors, following essentially the direction typified by the pioneering work of J. Sally. Our personal view of the subject is most visibly expressed by the presentation of Chapters 1 and 2 in which we discuss the use of the superficial elements and related devices. Basic techniques will be stressed with the aim of reproving recent results by using a more elementary approach. Over the past few years several papers have appeared which extend classical results on the theory of Hilbert functions to the case of filtered modules. The extension of the theory to the case of general filtrations on a module has one more important motivation. Namely, we have interesting applications to the study of graded algebras which are not associated to a filtration, in particular the Fiber cone and the Sally-module. We show here that each of these algebras fits into certain short exact sequences, together with algebras associated to filtrations. Hence one can study the Hilbert function and the depth of these algebras with the aid of the know-how we got in the case of a filtration.